Under the same reaction conditions, initial concentration of 1.386 mol ${\text{dm}}^{-3}$of a substance   becomes half in 40 seconds and 20 seconds through first order and zero order kinetics, respectively. Ratio $\left({\text{k}}_{\text{1}}{\text{/k}}_{\text{0}}\right)$of the rate constant for first order.

The values of rate constants for zero order and first order reaction, respectively

            , are given by the following equation:

${\text{k}}_{\text{0}}\text{\hspace{0.17em}=\hspace{0.17em}}\frac{{\text{A}}_{\text{0}}}{{\text{2\hspace{0.17em}×\hspace{0.17em}t}}_{\text{1/2}}}[{\text{where\hspace{0.17em}A}}_{\text{0}}{\text{\hspace{0.17em}=\hspace{0.17em}initial\hspace{0.17em}concetration,\hspace{0.17em}and\hspace{0.17em}t}}_{\text{1/2}}\text{\hspace{0.17em}=\hspace{0.17em}shalf\hspace{0.17em}}-\text{\hspace{0.17em}life\hspace{0.17em}period}]$

${\text{and\hspace{0.17em}k}}_{\text{1}}\text{\hspace{0.17em}=\hspace{0.17em}}\frac{\text{0.693}}{{\text{t}}_{\text{1/2}}}$        

  Substituting various given values, we get

   ${\text{k}}_{\text{0}}\text{\hspace{0.17em}=\hspace{0.17em}}\frac{{\text{1.386\hspace{0.17em}mol\hspace{0.17em}litre}}^{\text{-1}}}{\text{2\hspace{0.17em}×\hspace{0.17em}20\hspace{0.17em}sec}}\mathrm{...}\left(\text{i}\right)$

  ${\text{and\hspace{0.17em}k}}_{\text{1}}\text{\hspace{0.17em}=\hspace{0.17em}}\frac{\text{0.693}}{\text{40\hspace{0.17em}sec}}\mathrm{...}\left(\text{ii}\right)$

$\text{Dividing\hspace{0.17em}}\left(\text{ii}\right)\text{\hspace{0.17em}by\hspace{0.17em}}\left(\text{i}\right)\text{,\hspace{0.17em}we\hspace{0.17em}get}$

$\frac{{\text{k}}_{\text{1}}}{{\text{k}}_{\text{0}}}\text{\hspace{0.17em}=\hspace{0.17em}}\frac{\text{0.693}}{\text{40}}\text{\hspace{0.17em}×\hspace{0.17em}}\frac{\text{2\hspace{0.17em}×\hspace{0.17em}20}}{{\text{1.386\hspace{0.17em}mol\hspace{0.17em}dm}}^{\text{-3}}}\text{\hspace{0.17em}=\hspace{0.17em}}\frac{\text{0.693}}{\text{1.386}}\text{\hspace{0.17em}=\hspace{0.17em}0.5}$

${\text{=\hspace{0.17em}0.5\hspace{0.17em}mol}}^{\text{-1}}{\text{dm}}^{\text{3}}$

Thus the correct answer is (A).